ABSTRACT
Distance models for three-way proximity data, which consist of numerical values assigned to triples of objects that indicate their joint (lack of) homogeneity or resemblance, require a generalization of the usual distance concept defined on pairs of objects. An axiomatic framework is given for characterizing triadic dissimilarity, triadic similarity, and triadic distance, where the term triadic implies that each element of the triple is treated on an equal footing. Two kinds of distance models are studied in detail: the Minkowski-p or Mp model, which is based upon dyadic components and includes the perimeter model as an important special case, and several models based on presence-absence variables. They are shown to satisfy the tetrahedral inequality, a condition that is characteristic for the present axiomatization. Two monotonically convergent algorithms are described that find weighted least squares representations of three-way proximity data under the Euclidean M1 model and the Euclidean M2 model. To enable a scalefree evaluation of the quality of the fit, an additive decomposition of the sum of squares of the dissimilarities is derived. As illustrated in one of the examples, distance analysis of three-way, three-mode tables is possible by a suitable manipulation of the least squares weights.
